Pathways of Maximum Likelihood for Rare Events in Nonequilibrium Systems: Application to Nucleation in the Presence of Shear

Matthias Heymann, Eric Vanden-Eijnden
2008 Physical Review Letters  
Even in nonequilibrium systems, the mechanism of rare reactive events caused by small random noise is predictable because they occur with high probability via their maximum likelihood path (MLP). Here a geometric characterization of the MLP is given as the curve minimizing a certain functional under suitable constraints. A general purpose algorithm is also proposed to compute the MLP. This algorithm is applied to predict the pathway of transition in a bistable stochastic reaction-diffusion
more » ... ion in the presence of a shear flow, and to analyze how the shear intensity influences the mechanism and rate of the transition. The description of rare reactive events in nonequilibrium systems which lack detailed balance represents a difficult theoretical and computational challenge. Examples of such events include phase transitions, chemical reactions, biochemical switches, or regime changes in climate. Brute force simulation of these events by Monte Carlo [1] or direct simulation of Langevin equations [2] is difficult because of the huge disparity between the time step which must be used to perform the simulations and the time scale on which the rare events occur. Familiar concepts such as the minimum energy path which are often used to explain rare events are inappropriate because there is no energy landscape over which the system navigates. Methods to accelerate the simulation of these events [3] also typically fail in nonequilibrium situations. Despite recent theoretical [4] and numerical [5] advances to bypass these difficulties, many issues remain open. One aspect of rare events which can be exploited is that the pathway of these events is often predictable, even in nonequilibrium systems. This is the essence of large deviation theory [6] . When an improbable event occurs, the probability that it does so in any other way than the most likely one is very small because all these other ways are even much less probable. It then becomes important to identify the maximum likelihood path (MLP) of the rare event, i.e., the path which maximizes the event likelihood over all possible pathways and times this event can take to occur. The MLP explains the mechanism of the rare event, which is usually nontrivial and informative in complex systems. The MLP also permits to calculate other important quantities such as the rate of the event. In this Letter, we show that the MLP can be characterized geometrically as the directed curve minimizing a certain action functional, and we design an efficient general purpose algorithm to compute the MLP. We use this algorithm to study phase transitions in a Ginzburg-Landau-type model in the presence of shear [7] . In first approximation this model can be used as a crude approximation to describe, e.g., nuclea-tion of protein crystals, a problem which has received lots of attention recently because of its importance for the pharmaceutical industry [8] . We find the MLP in this model and estimate the rate of the transition in function of the shear intensity. We also show how to automatically identify the shear intensity which maximizes this rate. Part of the results presented here are justified rigorously in Ref. [9] .
doi:10.1103/physrevlett.100.140601 pmid:18518017 fatcat:glm3c6jyqnc2rjywoypvohs6sm